Domain Decomposition Learning Methods for Solving Elliptic Problems

摘要：

With recent advancements in computer hardware and software platforms, there has been a surge of interest in solving partial differential equations with deep learning-based methods, and the integration with domain decomposition strategies has attracted considerable attention owing to its enhanced representation and parallelization capacities of the network solution. While there are already several works that substitute the subproblem solver with neural networks for overlapping Schwarz methods, the non-overlapping counterpart has not been extensively explored because of the inaccurate flux estimation at interface that would propagate errors to neighbouring subdomains and eventually hinder the convergence of outer iterations. In this study, a novel learning approach for solving elliptic boundary value problems, i.e., the compensated deep Ritz method using neural network extension operators, is proposed to enable reliable flux transmission across subdomain interfaces, thereby allowing us to construct effective learning algorithms for realizing non-overlapping domain decomposition methods (DDMs) in the presence of erroneous interface conditions. Numerical experiments on a variety of elliptic problems, including regular and irregular interfaces, low and high dimensions, two and four subdomains, and smooth and high-contrast coefficients are carried out to validate the effectiveness of our proposed algorithms. 

报告人简介：

孙琪，2013年本科毕业于中国科学技术大学，随后在与北京计算科学研究中心的联合培养项目下获得计算数学博士学位，并在此期间赴美国哥伦比亚大学公派留学两年。2019 - 2021年在北京大学国际数学研究中心从事博士后工作，随后加入同济大学数学科学学院并担任助理教授职位。主要从事不确定性量化、区域分解方法、最优控制理论和机器学习的交叉领域研究，相关成果发表于JCP、CiCP、SINUM和NeurIPS 等国际知名期刊与会议。